Distribution of joint local and total size and of extension for avalanches in the Brownian force model

Mathieu Delorme, Pierre Le Doussal and Kay Jorg WieseCNRS-Laboratoire de Physique Theorique de l’Ecole Normale Superieure, 24 rue Lhomond, 75005 Paris, France

The Brownian force model (BFM) is a mean-field model for the local velocities during avalanches in elasticinterfaces of internal space dimension d, driven in a random medium. It is exactly solvable via a non-lineardifferential equation. We study avalanches following a kick, i.e. a step in the driving force. We first recall thecalculation of the distributions of the global size (total swept area) and of the local jump size for an arbitrary kickamplitude. We extend this calculation to the joint density of local and global sizes within a single avalanche, inthe limit of an infinitesimal kick. When the interface is driven by a single point we find new exponents τ0 = 5/3and τ = 7/4, depending on whether the force or the displacement is imposed. We show that the extension ofa “single avalanche” along one internal direction (i.e. the total length in d = 1) is finite and we calculate itsdistribution, following either a local or a global kick. In all cases it exhibits a divergence P (`) ∼ `−3 at small`. Most of our results are tested in a numerical simulation in dimension d = 1.

I. INTRODUCTION

In many physical systems the motion is not smooth, butproceeds by avalanches. This jerky motion is correlated overa broad range of space and time scales. Examples are mag-netic interfaces, fluid contact lines, crack fronts in fracture,strike-slip faults in geophysics and many more [1–3]. Thesesystems have been described using the model of an elastic in-terface slowly driven in a random medium. This model is im-portant for avalanches, both conceptually and in applications[4–6]. The full model of an interface of internal dimensiond, in presence of realistic short-ranged disorder is difficult totreat analytically, and requires methods such as the FunctionalRenormalization Group (FRG) [7–14].

A simpler version of the model, the so-called Brownianforce model (BFM) introduced in [10–13] is very interest-ing in several respects. First it is exactly solvable, and sev-eral avalanche observables have been calculated, as discussedbelow. Second, it was shown [11, 13] to be the appropriatemean-field theory for the space-time statistics of the velocityfield in a single avalanche for d-dimensional interfaces closeto the depinning transition for d ≥ duc with duc = 4 forshort ranged elasticity and duc = 2 for long-ranged elasticity.Remarkably, when considering the dynamics of the center ofmass of the interface, it reproduces the results of the simplerABBM model, a toy model for a single degree of freedom(particle), introduced long ago on a phenomenological basisto describe Barkhausen experiments (magnetic noise)[15, 16]and much studied since [1, 12, 17]. Last but not least, theBFM is the starting point for a calculation of avalanche ob-servables beyond mean-field, using the FRG in a systematicexpansion in duc − d [11, 13].

The key property which makes the BFM (and the ABBM)model solvable is that the disorder is taken to be a Brown-ian random force landscape. Since it can be shown that un-der monotonous forward driving the interface always movesforward (Middleton’s theorem [18]), the resulting equation ofmotion for the velocity field is Markovian, and amenable toexact methods.

Despite being exactly solvable, the explicit calculation ofavalanche observables in the BFM requires solving a non-

linear instanton equation and performing Laplace inversions,which is not always an easy task. Global avalanche properties,such as the probability distribution function (PDF) of globalsize S, of duration, and of velocity have been obtained for ar-bitrary driving. Detailed space time properties however aremore difficult. In Ref. [13] a finite wave-vector observablewas calculated, demonstrating an asymetry in the temporalshape. Although the distribution of local avalanche sizes Srhas been obtained in some instances, this is not the case forthe distribution of the spatial extension ` of an avalanche, i.e.the range of points which move during an avalanche, an im-portant observable accessible in experiments. Note that eventhe fact that an avalanche has a finite extent, instead of anexponentially decaying tail in its spatial extension is a non-trivial result, which up to now was only proven for very largeavalanches in the BFM [19].

x

u

rl

SrS

uxt=∞uxt=0

FIG. 1. An avalanche in d = 1.

arX

iv:1

601.

0494

0v1

[co

nd-m

at.d

is-n

n] 1

9 Ja

n 20

16

2

Driving protocol Observable Exponent

any force kick global size S τ = 3/2

uniform force kick local size S0 τ0 = 4/3

uniform force kick S0 at fixed S τ0 = 2/3

localized force kick local size S0 τ0 = 5/3

local displacement imposed global size S τ = 7/4

any force kick extension ` κ = 3

TABLE I. Summary of small-scale exponents for different distribu-tions in the Brownian-Force Model, depending on the observable andthe driving protocol.

The aim of this paper is to calculate further observables forthe BFM which contain information about local properties,such as the joint density of global and local avalanches, andthe distribution of extensions. We consider various protocols,where the interface is either driven uniformly in space orat a single point; in the latter case we identify new criticalexponents. We study avalanches following a kick, i.e. a stepin the driving force.

The article is structured as follows: In section II we re-call the definition of the BFM and of the main avalanche ob-servables, together with the general method to obtain themfrom the instanton equation. Section III starts by recalling thecalculation of the distributions of the global size (total sweptarea) S and of the local jump size Sr of an avalanche, for anarbitrary kick amplitude. In Section III C we extend this cal-culation to the joint density ρ(Sr, S) of local and global sizefor single avalanches, i.e. in the limit of an infinitesimal kick.In Section IV we study the case of an interface driven at a sin-gle point. When the force at this point is imposed, we find anew exponent τ0 = 5/3 for the PDF of the local jump S0 atthat point. When the local displacement is imposed, we finda new exponent τ = 7/4 for the PDF of the global size S. InSection V we show that the extension ` of a single avalanchealong one internal direction (i.e. the total length in d = 1) isfinite; we calculate its distribution, following either a local ora global kick. In all cases it exhibits a divergence P (`) ∼ `−3

at small `, with the same prefactor. All these exponents can befound in Table I. Finally, in Section VI we study the positionof the interface, which is a non-stationary process. We explainhow the Larkin and BFM roughness exponents emerge fromthe dynamics. Most of our results are tested in a numericalsimulation of the equation of motion in d = 1.

The technical parts of the calculations are presented in Ap-pendices A to J, together with general material about Airy,Weierstrass and Elliptic functions. A short presentation of thenumerical methods is also included.

Finally note a complementary recent study of the BFM,where the joint PDF of the local avalanche size at all pointswas obtained. From that, the spatial shape of an avalanche inthe limit of large aspect ratio S/`4 was derived [19].

II. AVALANCHE OBSERVABLES OF THE BFM

A. The Brownian Force Model

In this paper, we study the Brownian Force Model (BFM) inspace dimension d, defined as the stochastic differential equa-tion (in the Ito sense) :

η∂tuxt = ∇2xuxt +

√2σuxt ξxt +m2(wxt − uxt) . (1)

This equation models the overdamped time evolution, withfriction η, of the velocity field uxt ≥ 0 of an interface withinternal coordinate x ∈ Rd; the space-time dependence is de-noted by indices u(x, t) ≡ uxt. It is the sum of three contri-butions:

• short-ranged elastic interactions,

• stochastic contributions from a disordered medium,where ξ is a unit Gaussian white noise (both in x andt) :

ξxtξx′t′ = δd(x− x′) δ(t− t′), (2)

• a confining quadratic potential of curvaturem, centeredat wxt, acting as a driving.

The driving velocity is chosen positive, wxt ≥ 0, a necessarycondition for the model to be well defined, as it implies thatuxt ≥ 0 at all t > 0 if uxt=0 ≥ 0.

Equation (1), taken here as a definition, can also be derivedfrom the equation of motion of an elastic interface, parameter-ized by a position field (displacement field) uxt in a quenchedrandom force field F (u, x),

η∂tuxt = ∇2xuxt + F (uxt, x) +m2(wxt − uxt) . (3)

The random force field is a collection of independent one-sided Brownian motions in the u direction with correlator

F (u, x)F (u′, x′) = 2σδd(x− x′) min(u, u′) . (4)

Taking the temporal derivative ∂t of Eq. (3), and assumingforward motion of the interface, one obtains Eq. (1) for thevelocity variable ∂tuxt ≡ uxt (we use indifferently ∂t or a dotto denote time derivatives). The fact that the equation for thevelocity is Markovian even for a quenched disorder is remark-able and results from the properties of the Brownian motion.

Details of the correspondence are given in [12, 13] wheresubtle aspects of the position theory, and its links to the mean-field theory of realistic models of interfaces in short-rangeddisorder via the Functional Renormalisation Group (FRG) arediscussed. In the last section of this paper we will mentionsome properties of the position theory of the Brownian forcemodel.

B. Avalanches observables and scaling

The BFM (1) allows to study the statistics of avalanchesas the dynamical response of the interface to a change in the

3

driving. We consider solutions of (1) as a response to a drivingof the form

wxt = δwx δ(t) , δwx ≥ 0 , δw = L−d∫x

δwx > 0 . (5)

The initial condition is

uxt=0 = 0 . (6)

This solution describes an avalanche which starts at time t = 0and ends when uxt = 0 for all x. The time at which theavalanche ends, also called avalanche duration, was studied in[20] and its distribution given in various situations.

Within the description (3), i.e. in the position theory, itcorresponds to an interface pinned, i.e. at rest in a metastablestate at t < 0, it is submitted at t = 0 to a jump in thetotal applied force m2δw. More precisely, the center of theconfining potential jumps at t = 0 from wx (where it wasfor t < 0) to wxt=0+ = wx + δwx (where it stays for allt > 0). As a consequence, the interface moves forward (sinceδwx ≥ 0) up to a new metastable state. This is representedin figure 1, where uxt=0 is the initial metastable state anduxt=∞ is the new metastable state at the end of the avalanche.In fact, as we will see from the distribution of avalanchedurations, the new metastable state is reached almost surelyin a finite time. For details on these metastable states and thesystem’s preparation see [12, 13].

We now discuss the avalanche observables at the center ofthis paper. They can be computed from the solution of (1)given (5) and (6); they are represented in figure 1 for a morevisual definition in the case d = 1.

• Global size of the avalanche:

S =

∫x∈Rd

∫ ∞0

utx dt . (7)

This is the total area swept by the interface during theavalanche.

• Local size of the avalanche:

Sr = m−1

∫x∈r×Rd−1

∫ ∞0

utx dt . (8)

This is the size of the avalanche localized on a hy-perplane, where one of the internal coordinates is r;the factor m−1 allows to express S and Sr using thesame units (see below). In d = 1 this yields Sr =m−1

∫∞0utr dt, i.e. the transversal jump at the point r

of the interface. For d > 1 the variable r is still one-dimensional, and Sr the total displacement in a hyper-plane of the interface.

• Avalanche extension:

For d = 1, the extension (denoted `) of an avalanche isthe lenght of the part of the interface which (strictly)moves during the avalanche. The generalisation to

x

yu

l

S = 0

Supp

r1 r1

r2 S > 0r2

FIG. 2. An avalanche in d = 2; the transverse direction is orthogonalto the plane of the figure and the colored zone corresponds to thesupport of the avalanche.

avalanches of a d-dimensional interface is done with thedefinition

` =

∫ ∞−∞

dr θ(Sr > 0) , (9)

where θ is the Heaviside function. Note that even for ad-dimensional interface, the extension ` is a unidimen-sional observable (cf. figure 2).

Note that

Sr > 0 ⇔ Supp⋂r × Rd−1 6= ∅ (10)

where Supp denotes all the points of the interface moving dur-ing an avalanche (i.e. its support).

We use natural scales (or units) to switch to dimensionlessexpressions, both for the (local and global) avalanche size Sm,as for the time τm expressed as

Sm =σ

m4, τm =

η

m2. (11)

The extension, a length in the internal direction of the inter-face, is expressed in units ofm−1. This is equivalent to settingm = σ = η = 1. All expressions below, except explicit men-tion, are expressed in these units.

While Sm is the large-size cutoff for avalanches, there isgenerically also a small-scale cutoff. As in the BFM the dis-order is scale-invariant (by contrast with more realistic mod-els with short-ranged smooth disorder), it is the increment inthe driving δw which sets the small-scale cutoff for the localand global size of avalanches. They scale as min(S) ∼ δw2

(global size) and min(Sr) ∼ δw3 (local size).

Massless limit

There are cases of interest where the mass m → 0. Thiscan be defined from the equations of motion (1) and (3) with

4

the changes

m2(wxt − uxt)→ fxt

m2(wxt − uxt)→ fxt .(12)

In that case it is natural to consider driving with a given forcefxt, rather than by a parabola. The definition of the observ-ables is the same except that the factor of m−1 is not addedin the definition (8). To bring σ and η to unity, we then definetime in units of η and displacements u in units of σ. The re-sults will still have an unfixed dimension of length. In someof them, the system size L leads to dimensionless quantities(it also acts as a cutoff for large sizes, although we will notuse this explicitly).

C. Generating functions and instanton equation

To compute the distribution of the observables presentedabove, we use a result from [12, 13] which allows us to ex-press the average over the disorder of generating functions(Laplace transforms) of uxt, solution of (1). In dimensionlessunits, this result reads

G[λxt] =

⟨exp

(∫xt

λxtuxt

)⟩= e

∫xtwxtuxt . (13)

Here 〈· · · 〉 denotes the average over disorder. u is a solutionof the differential equation (called instanton equation)

∂2xu+ ∂tu− u+ u2 = −λxt . (14)

Since avalanche observables that we consider are integrals ofthe velocity field over all times (c.f. observable definitionsabove), the sources λxt we need in (13) are time independent.Thus we only need to solve the space dependent, but time in-dependent, instanton equation

u′′x − ux + u2x = −λx . (15)

The prime denotes derivative w.r.t x. In the massless casediscussed above, the term −ux is absent, all other terms areidentical.

The global avalanche size implies a uniform source in theinstanton equation: λx = λ, while the local size implies alocalized source λx = λδ1(x). To obtain information on theextension of avalanches, we need to consider a source local-ized at two different points in space, λx = λ1δ(x − r1) +λ2δ(x− r2).

This instanton approach, which derives from the Martin-Siggia-Rose formulation of (1), allows us to compute exactlydisorder averaged observables for any form of driving, bysolving a “simple” ordinary differential equation, which de-pends on the observable we want to compute, i.e. on λxt, butnot on the form of the driving wxt. For a derivation of (13)and (14) we refer to [12].

10-3 10-2 10-1 100 101

10-5

10-4

10-3

10-2

10-1

100

101

102

103 Theoretical distribution Pδw(S)

Numerical simulation

FIG. 3. Green histogram : global avalanche-size distribution from adirect numerical simulation of a discretized version of Eq. (1) withparameters : N = 1024,m = 0.01, df = m2δw = 1 and dt =0.05. Red line : theoretical result given in Eq. (16). For details aboutthe simulation see appendix H.

III. DISTRIBUTION OF AVALANCHE SIZE

A. Global size

As defined in (7) the global size of an avalanche is the totalarea swept by the interface. Its PDF was calculated in [11–13]and reads, in dimensionless units,

Pδw(S) =δw

2√πS

32

e−(S−δw)2

4S . (16)

Here δw = Ldδw. This result does not depend on the spatialform of the driving (it can be localized, uniform, or anythingin between), as long as it is applied as a force on the interface.Driving by imposing a specific displacement at one point ofthe interface is another interesting case that leads to a differentbehavior, see Section IV B.

We can test this against a direct numerical simulation of theequation of motion (1). There is excellent agreement over 5decades, with no fitting parameter, see Fig. 3.

Avalanches have the property of infinite divisiblity, i.e. theyare a Levy process. This can be written as an equality in dis-tribution, i.e. for probabilities,

Pδw1∗ Pδw2

d= Pδw1+δw2

. (17)

It implies that we can extract from the probability distribution(16) the single avalanche density per unit δw that we denoteρ(S) and which is defined as

Pδw(S) 'δw1

δw ρ(S) . (18)

This avalanche density contains the same information as thefull distribution (16); its expression is

ρ(S) =Ld

2√πS

32

e−S4 ∼ S−τ . (19)

5

It is proportional to the system volume since avalanches occuranywhere along the interface. It defines the avalanche expo-nent τ = 3

2 for the BFM. Due to the divergence when S → 0it is not normalizable (it is not a PDF), but as the interface fol-lows on average the confining parabola, it has the followingproperty ∫ ∞

0

dS Sρ(S) = Ld . (20)

In this picture, typical, i.e. almost all avalanches are of van-ishing size, S ≈ 0, or more precisely S ≤ δw2, but momentsof avalanches are dominated by non-typical large avalanches(of order Sm).

B. Local size

We now investigate the distribution of local size Sr as de-fined in Eq. (8). We have to specify the form of the kick; westart with one uniform (in x): δwx = δw for all x ∈ R. In thiscase the system is translationnaly invariant, and we can chooser = 0, as any local size will have the same distribution.

The distribution of S0 is obtained by solving Eq. (15)with the source λx = λδ(x), and then computing theinverse Laplace transform with respect to λ of G(λ) =exp(δw

∫xuλ), where uλ is the instanton solution (depend-

ing on λ). This has been done in [13]; the final result is

Pδw(S0) =2× 3

13

S430

e6δwδwAi

((3

S0

)13

(S0 + 2δw)

)

'δw1 δw2Ld−1

πS0K 1

3

(2S0√

3

).

(21)

Here δw = Ld−1δw, Ai is the Airy function, and K the Besselfunction. We use that Ai(x) = 1

π

√x3K1/3( 2

3x3/2) for x > 0.

This distribution has again the property of infinite divisibil-ity, which is far from obvious on the final results but, can bechecked numerically.

The small-δw limit defines the density per unit δw of thelocal sizes of a “single avalanche”, which is given by

ρ(S0) =2Ld−1

πS0K 1

3

(2S0√

3

)'S01 L

d−16√

3 Γ(1/3)

πS04/3

∼ S−τφ0 .

(22)

Its small-size behavior defines the local size exponent τφ = 43

for the BFM.The distribution (21), or the density (22), can be compared

to the results of direct numerical simulations of the BFM, andthe agreement is very good over 7 decades, without any fittingparameter, c.f. Fig. 4.

Another interesting property is that the tail of large localsizes behaves as ρ(S0) 'S01 S

−3/20 e−2S0/

√3, i.e. with the

same power-law exponent in the pre-exponential factor as theglobal size.

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

105

106Theoretical distribution Pδw(S0 )

Numerical simulation

FIG. 4. Green histogramm: Local avalanche-size distribution froma direct numerical simulation of a discretized version of (1)with pa-rameters N = 1024,m = 0.01, df = m2δw = 1, and dt = 0.05.Red line: the theoretical result given in Eq. (21). For details aboutthe simulation see appendix H.

C. Joint global and local size

We now extend these results with a new calculation of thejoint density of local and global sizes. Consider Pδw(S0, S),the joint PDF of local size S0 and global size S, following auniform kick δw. For arbitrary δw it does not admit a simpleexplicit form (see Appendix D). We thus again consider the“single avalanche” limit δw → 0. It defines the joint den-sity ρ(S, S0), via Pδw(S0, S) ' δw ρ(S0, S), which we nowcalculate. Equivalently one can consider the conditional prob-ability Pδw(S0|S) of the local size, given that the global sizeis S. In the limit δw → 0 these two objects are related by

P0+(S0|S) =ρ(S0, S)

ρ(S), (23)

where ρ(S) is given in Eq. (19); the two factors of δw can-cel. For simplicity we discuss the result for P0+(S0|S). Whileboth ρ(S) and ρ(S0, S) are not probabilities, i.e. they cannotbe normalized to one, we will show that the conditional prob-ability P0+(S0|S) is well-defined, and normalized to unity.

A natural decomposition of this conditional PDF is

P0+(S0|S) = P0+(S0|S) + δ(S0)

(1−

∫u>0

P0+(u|S)

).

(24)The first term is the smooth part defined for S0 > 0 whichcomes from the avalanches containing the point r = 0. Thesecond term arises from all avalanches which do not containthe point r = 0. This term contains a substraction so that thetotal probability is normalized to unity,

∫S0P0+(S0|S) = 1,

as it should be.The smooth part is calculated using the instanton-equation

approach. The details are given Appendix D. The final result

6

10-6 10-5 10-4 10-3 10-2 10-1 100

10-2

10-1

100

101

102

103

104

105

Numerical distribution with S>0.001

Numerical distribution with S>0.01

Numerical distribution with S>0.1

Theoretical distribution (δw=0+ )

FIG. 5. Distribution of α, defined in Eq. (26), from numerical sim-ulations (N = 1024,m = 0.02, δw = 10, dt = 0.01). This iscompared to the theoretical prediction (33). Keeping only large-sizeavalanches, this converges (without any adjustable parameter) to theδw = 0+ result.

takes the scaling form

P0+(S0|S) =1

L

4× 323

S230

e−23α

3[αAi

(α2)− Ai′

(α2)]

(25)

with

α :=3

23S

430

S. (26)

The factor 1/L is natural since only a fraction of order 1/L ofavalanches contains the point r = 0. As written, this smoothpart is not normalized. Its integral is equal to the probability pthat the point S0 has moved (i.e. S0 > 0) during an avalanche,for which we find

p :=

∫ ∞0

dS0 P0+(S0|S) =S

14

L

3Γ(

14

)√π

. (27)

The scaling of this probability with size shows that in a singleavalanche only a finite portion of the interface is moving. Ifwe assume statistical translational invariance we deduce that

p = 〈`〉S/L , (28)

where ` is the extension defined in (9), and 〈`〉S its mean valueconditioned to the global size S. Hence we deduce that

〈`〉S =3Γ(

14

)√π

S14 . (29)

In the following sections we will in fact calculate the PDF ofthe extension `.

By dividing by p, we can now define a genuine normalizedPDF for S0, P0+(S0|S), conditioned to both S and S0 > 0, sothat the decomposition (24) becomes

P0+(S0|S) = p P0+(S0|S) + δ(S0)(1− p) . (30)

Explicitly

P0+(S0|S) =4√πe−

23α

3

313 Γ(

14

)S

230 S

14

[αAi

(α2)− Ai′

(α2)], (31)

with α defined in Eq. (26). It is now normalized to unity,∫S0>0

P0+(S0|S) = 1. One sees that the typical local sizescales as S0 ∼ S3/4. Computing the first moment we find itsconditional average to be 〈S0〉S,S0>0 =

√π

3Γ(1/4)S3/4. Its PDF

has two limiting behaviors,

P0+(S0|S) '

e−

12S40

S3

Γ( 54 )S

34

for S0 S34

√π

323 Γ( 1

3 )Γ( 54 )S

230 S

14

for S0 S34 .

(32)

The first one shows that the probability of avalanches whichare “peaked” at r = 0 decays very fast. The second showsan integrable divergence at small S0 with an exponent 2/3.Comparing, for instance, with the behavior of the local sizedensity (22), we see that conditioning on S yields a ratherdifferent behavior and exponent.

It is interesting to note that changing variables in Eq. (31)from S0 to α, defined in (26), gives

P0+(α|S) =

√3πe−

23α

3

Γ(

14

)α

34

[αAi

(α2)− Ai′

(α2)], (33)

which is now independant of S, and thus easier to test numeri-cally as it does not require any conditionning. Figure 5 showsthe agreement of these predictions with numerical simula-tions, in the limit of large S which is equivalent to δw = 0+

as used in the theoretical derivation.

D. Scaling exponents

Let us now discuss the various exponents obtained untilnow. They are consistent with the usual scaling arguments forinterfaces. If an avalanche has an extension of order ` (in thecodirection of the hyperplane over which the local size is cal-culated), the transverse displacement scales as u ∼ `ζ . Herethe roughness exponent ζ for the BFM with SR elasticity is

ζBFM = 4− d . (34)

The avalanche exponent for the global size follows theNarayan-Fisher (NF) prediction [8]

τ = 2− 2

d+ ζ

BFM−−→ 3

2. (35)

The global size then scales as S ∼ `d+ζ , since all d internaldirections are equivalent, and the transverse response scaleswith the roughness exponent ζ. In turn this gives ` ∼ S

1d+ζ .

In the BFM with SR elasticity this leads to ` ∼ S1/4 as foundabove.

7

Similarly, the local size, defined here as the avalanche sizeinside a dφ-dimensionel subspace, is S0 ∼ `dφ+ζ , leadingto a generalized NF value τφ = 2 − 2

dφ+ζ . In the BFM wehave focused on the case dφ = d − 1 (i.e. the subspace is anhyperplane), hence dφ + ζ = 3 and the local size exponentbecomes τφ = 4/3. It also implies S0 ∼ `3, hence S0 ∼ S3/4

as found above.

IV. DRIVING AT A POINT: AVALANCHE SIZES

Here we briefly study avalanche sizes for an interfacedriven only in a small region of space, e.g. at a point. Thereare two main cases:

• the local force on the point is imposed, which in ourframework means to consider a local kick δwx =δw δ(x). In the massless setting it amounts to usefx = δf δ(x),

• the displacement ux=0,t of one point of the interface isimposed.

As we now see this leads to different universality classesand exponents.

A. Imposed local force

Consider an avalanche following a local kick at x = 0, i.e.δwx = δw0δ(x).

In the BFM the distribution of the global size of anavalanche does not depend on whether the kick is local inspace or not. One still obtains [13] the global-size distribu-tion as given in Eq. (16) with δw =

∫xδwx = δw0.

The distribution of the local size at the point of the kick ismore interesting. The calculation is performed in AppendixC 2. For simplicity we restrict to d = 1, the general case canbe obtained as above by inserting factors of Ld−1. The full re-sult for the PDF, Pδw0

(S0), is given in (C7) and is bulky. In thelimit δw0 → 0 it simplifies. Noting Pδw0

(S0) ' δw0ρ(S0),the corresponding local-size density becomes

ρ(S0) = − 1

31/3S5/30

Ai′(

31/3S2/30

). (36)

At small S0, or equivalently in the massless limit at fixedδf0 = m2δw0, it diverges as

ρ(S0) 'S01

S−5/30

32/3Γ(1/3)∼ S−τ0,loc.driv.

0 . (37)

This leads to a new avalanche exponent

τ0,loc.driv. =5

3. (38)

The cutoff at small size is given by the driving, S0 ∼ δw3/20 .

At large S0 the PDF is cut by the scale Sm ≡ 1 and decays as

ρ(S0) 'S01

S−3/20

2√π31/4

e−2S0/√

3 . (39)

B. Imposed displacement at a point

We analyze the problem in the massless case. To imposethe displacement at point x = 0 we replace in the equationof motion (1) and (3), m2 → m2δ(x). Hence there is noglobal mass, but a local one to drive the interface at a point.To impose the displacement, we consider the limit m2 →∞.In that limit ux=0,t = w0,t, and the local size of the avalancheS0 is equal to δw0.

While the local size S0 is fixed by the driving, we can calcu-late the distribution of global sizes. It is obtained in AppendixE using an instanton equation with a Dirac mass term. It canbe mapped onto the same instanton equation as studied for thejoint PDF of local and global sizes. The Laplace-transform ofthe result for the PDF is given in Eq. (E6). Its small-drivinglimit, i.e. the density, is

ρ(S) =

√3

Γ(1/4)S7/4∼ S−τloc.driv. (40)

with a distinct exponent

τloc.driv. =7

4. (41)

V. DISTRIBUTION OF AVALANCHE EXTENSIONS

In this section we study the distribution of avalanche ex-tensions. In the BFM they can be calculated analytically. Westart by recalling standard scaling arguments.

A. Scaling arguments for the distribution of extensions

As mentioned in the last section, we expect that the globalsize S and the extension ` of avalanches are related by thescaling relation

S ∼ `d+ζ (42)

in the region of small avalanches S Sm (in dimensionfullunits). From the definition of the avalanche-size exponent

P (S) ∼ S−τ (43)

and using the change of variables P (S)dS = P (`)d` we find

P (`) ∼ `−κ with κ = 1 + (τ − 1)(d+ ζ) . (44)

Using the value for τ from the NF relation (35) we obtain

τ = 2− 2

d+ ζ. (45)

For SR elasticity, this yields

κ = d+ ζ − 1 . (46)

The prediction for the BFM is that ζBFM = 4−d and τBFM =3/2, which leads to

κBFM = 3 (47)

8

in all dimensions. We will now check this prediction from thescaling relations with exact calculations on the BFM model ind = 1.

B. Instanton equation for two local sizes

If we want to investigate the joint distribution of two lo-cal sizes at points r1 and r2, we need to solve the instantonequation with two local sources,

u′′x − ux + u2x = −λ1δ(x− r1)− λ2δ(x− r2) . (48)

This solution is difficult to obtain for general values of λ1 andλ2. Nevertheless λ1,2 → −∞ is an interesting solvable limit,and sufficient to compute the extension distribution. Let usdenote by ur1,r2(x) a solution of Eq. (48) with r1 < r2 in thislimit λ1,2 → −∞. It allows to express the probability thattwo local sizes in an avalanche following an arbitrary kickδwx equal 0,

Pδwx(Sr1 = 0, Sr2 = 0)

= exp

(∫x∈Rd

δwx ur1,r2(x)

).

(49)

We further restrict for simplicity to the massless case, i.e.without the linear term ux in Eq. (48). One easily sees fromthe latter equation that ur1,r2 takes the scaling form

ur1,r2(x) =1

(r1 − r2)2f

(2x− r1 − r2

2(r2 − r1)

). (50)

The function f(x) is solution of

f ′′(x) + f(x)2 = 0 . (51)

It diverges at x = ± 12 , vanishes at x → ±∞ and is negative

everywhere: f(x) ≤ 0. As δwx ≥ 0, the latter is a necessarycondition s.t. the probability (49) is bounded by one.

In the interval x ∈]− 12 ,

12 [, the scaling function f(x) can be

expressed in terms of the Weierstrass P-function, see (I15),

f(x) = −6P(x+

1

2; g2 = 0; g3 =

Γ(1/3)18

(2π)6

). (52)

The value of g3 > 0 is consistent with the required period2Ω = 1, see (I12). Note from Appendix I that there is another

solution of the form (52) with g3 = −(

2√π Γ(1/3)

413 Γ(5/6)

)6

< 0

which violates the condition f(x) ≤ 0, hence is discarded.For |x| ≥ 1/2, the function f(x) reads

f(x) = − 6

(|x| − 1/2)2. (53)

One property of the solution ur1,r2(x) is that it diverges as∼ (x− r1,2)−2 when x ≈ r1,2. There are thus two cases:

(i) - the driving δwx is non-zero at one of these points,or vanishes too slowly near this point (e.g. only linearly or

slower). Then the integral in (49) is not convergent, equal to−∞, which implies

Pδwx(Sr1 = 0, Sr2 = 0) = 0 .

This means that the avalanche contains surely at least one ofthe points r1 or r2.

(ii) - If δwx vanishes fast enough, for example if δwx is lo-calised away from x = ±r1,2 (e.g δwx = δwδ(x−y) for somey ∈ R\r1, r2), the probablity (49) becomes non trivial.

C. Avalanche extension with a local kick

We now consider a local kick centered at x = 0, i.e. wx =δw0 δ(x). If further 0 < r1 < r2, then

Pδw0(Sr1 = 0, Sr2 = 0) = Pδw0(Sr1 = 0) . (54)

This comes from the fact that in the interval x ∈ [−∞, r1], thesolution ur1,r2(x) is identical to the instanton solution withonly one infinite source at r1 (in other word, it does not “feel”the source in r2). This shows for instance that the support ofthe avalanche is larger or equal than the set of points wherethe driving is non-zero.

This property also shows that avalanches are connected, i.e.it is impossible to draw a plane where the interface did notmove between two moving parts of the interface. As a func-tion of r (which is one-dimensional), the support (i.e. the setof points where Sr > 0) of an avalanche following a localkick at x = 0 must be an interval. Since this interval con-tains x = 0 we will write it as [−`1, `2] with `1 > 0 and`2 > 0. This allows to define the extension of an avalanche as` = `1 + `2.

To calculate the joint PDF of `1 and `2 for a kick at x = 0we consider (49) with r1 = −x1 < 0 < r2 = x2. Us-ing the previous results about the instanton equation with twosources, and the fact that the interface model is translation-aly invariant, we obtain the joint cumulative distribution for`1 > 0 and `2 > 0:

Fδw0(x1, x2) := Pδw0

(`1 < x1, `2 < x2) . (55)

It can, for any x1, x2 > 0, be expressed in terms of the func-tion f obtained in the preceding section,

Fδw0(x1, x2) = Pδw0

(Sr1 = 0, Sr2 = 0)

= exp

(∫x

δw0δ(x) u−x1,x2(x)

)= e

δw01

(x1+x2)2f(− x2−x1

2(x1+x2)

) . (56)

Since the argument of f is within the interval ] − 12 ,

12 [ we

must use the expression (52).From this one can obtain several results. First taking x2 →∞ one obtains the PDF of `1 alone,

Pδw (`1) =12δw

`31e−δw 6

`21 . (57)

9

- 0.4 - 0.2 0.2 0.4

20

40

60

80

100

FIG. 6. Decay amplitude R(k) as a function of the aspect ratio kinvolved in the joint density of ` and k, and defined in Eqs. (60) and(61).

A similar result holds for `2.In principle, one can now obtain the distribution of

avalanches extensions

Pδw0(`) =

∫ ∞0

d`1

∫ ∞0

d`2 δ(`− `1 − `2)∂`1∂`2Fδw0(`1, `2)

(58)It has a rather complicated expression. Let us define in addi-tion to the total length, the aspect ratio

k =`1 − `2

2(`1 + `2), −1

2< k <

1

2. (59)

Using a change of variables, we obtain the joint density oftotal extension and aspect ratio in the limit δw0 → 0,

ρ (`, k) := limδw0→0

1

δw0Pδw0

(`, k) =R(k)

`3, (60)

R(k) := 6f(k) + 6kf ′(k) +

(k2 − 1

4

)f ′′(k) . (61)

The function f(x) was defined in Eq. (52). While the prob-ability as a function of ` decays as `−3, the dependenceon the aspect ratio is more complicated and plotted in fig-ure 6. Note that in this expression f(k) can be replaced byfreg(k) := f(k) + 6

(k+ 12 )2

+ 6(k− 1

2 )2, which is a regular func-

tion of k, vanishing at k = ± 12 . Integration over k gives

ρ (`) =B

`3with (62)

B = 24 + 2

∫ 1/2

−1/2

freg(k) = 8√

3π . (63)

D. Avalanche extension with a uniform kick

If a kick extends over the whole system, as e.g. a uniformkick δwx = δw, the avalanche will have almost surely aninfinite extension as the local size is non-zero everywhere,

Pδw (Sr = 0) = 0 for any r ∈ R . (64)

100 101

10-5

10-4

10-3

10-2

10-1

100

101

Small scale power lawLarge scale asymptoticTheoretical density distributionNumerical simulation

FIG. 7. The distribution of extensions ρ(`), as obtained from the el-liptic integrals (F7) and (F8) (black line). The (straight) green dottedline is the small-` asymptotics (69), whereas the (curved) red dottedline is the large-` asymptotics (71). The numerical simulation (greenhistogram) is cut at small scale due to discretization effects.

However, in the limit of a small δw which is also the limit of a“single avalanche”, we can recover the result for the distribu-tion of extensions. This is consistent with the idea that “sin-gle avalanches” do not depend on the way they are triggered.These calculations allow to obtain the extension distributionwithout solving explicitly the instanton equation. (The use ofelliptic integrals is in fact equivalent to the use of Weierstrassfunctions as solutions of the instanton equation, c.f. AppendixI).

We now focus on the following ratio of generating functions

〈eλ1s0+λ2sr 〉〈eλ1s0〉〈eλ2sr 〉

(65)

in the limit λ1, λ2 → −∞. It compares the probability thatboth local sizes s0 := S0 and sr := Sr are simultaneously 0to the product of the two probabilties that each one is 0.

We can express this ratio, using the instanton-equation ap-proach, as

limλ1,λ2→−∞

〈eλ1s0+λ2sr 〉〈eλ1s0〉〈eλ2sr 〉

= exp

(∫x

δwx

[ur(x)− u∞(x)− u∞(x− r)

]) (66)

where ur := ur1=0,r2=r. We denote by u∞ := ur1=0,r2=∞,the solution of the instanton equation with one source at r = 0and the other one at infinity. It is the same as the solution foronly one source in r = 0. The above expression is valid forany form of driving δwx.

We can now specify to the case of small and uniform driv-ing δwx = δw; the quantity of interest is then

Z(r) =

∫x

ur(x)− u∞(x)− u∞(x− r) . (67)

10

While ur(x) is not integrable, Z(r) is well defined as the twou∞ terms cancel precisely the two non-integrable poles lo-cated at x = 0 and x = r.

Using that ur is a solution of Eq. (48), we can obtain anexpression of Z(r) as an elliptic integral, see Appendix F fordetails of the calculation. The formulas written there are forthe massive case, but only allow to get an implicit expressionfor Z(r). They however allow us to extract the small-scalebehavior of the avalanche-extension distribution (equivalentlythe massless limit). For small r, the behavior of Z(r) is

Z(r) ' 4√

3π

r. (68)

To understand the connection with the avalanche extension,we need to get back to the interpretation of (65). Now that wehave specified the kick to be uniform, the two averages of thedenominator are independant of r, and act only as a normal-ization constants. The numerator, in the limit of λ1,2 → −∞,is the probability that both s0 and sr are simultaneously equalto 0. Deriving this two times w.r.t. r (which lets the denom-inator invariant) gives the probability that the avalanche startin x = 0 and end in x = r. Dividing by δw and taking thelimit1 δw → 0 , we obtain the extension density in the limitof a single avalanche as

ρ(`) =1

δw∂2reδwZ(r)|δw=0+,r=` (69)

= ∂2r Z(r)

∣∣∣r=`' B`−3 when `→ 0

with

B = 8√

3π . (70)

We recover here the `−3 divergence for small ` of the exten-sion of avalanches. Note that this calculation gives exactlythe same prefactor as in Eq. (62), which confirms that we arestudying the same object, namely a “single avalanche”.

Finally, in the massive case, one can also compute the tailof the extension distribution, resulting into (see Appendix F)

ρ(`) ' 72 `e−` when `→∞ . (71)

VI. NON-STATIONNARY DYNAMICS IN THE BFM

The easiest way to construct a position theory equiva-lent to the BFM model define in Eq. (1) is to consider thenon-stationnary evolution of an elastic line in some specificquenched disorder,

η∂tuxt = ∇2xuxt + F (uxt, x) +m2(wxt − uxt) . (72)

1 Note that the denominators can then be set to unity. There is no ambiguitysince the calculation could be performed first at finite but large λi, andsetting δw to zero after taking the derivative and dividing by δw, and onlyat the end taking the limit of infinite λi.

Here the disorder has the correlations of independent one-sided Brownian motion

F (u, x)F (u′, x′) = 2σδd(x− x′) min(u, u′) . (73)

Consider the initial condition uxt=0 = 0. We can then com-pute the correlation function of the position

uxt =

∫ t

0

uxs ds

for a uniform driving wt = vt θ(t), starting at t = 0. Thecalculation is sketched in Appendix J. In dimensionless unitsand in Fourier space, the result reads

〈uqtu−qt〉c = v

[2q2(t− 1) + 2t− 5

(q2 + 1)3 − 4e−(q2+1)t

q2 (q2 + 1)3

+4e−t

q2 (2q2 + 1)+

e−2(q2+1)t

(q2 + 1)3

(2q2 + 1)

]. (74)

At large times, the displacement correlations behave as(restoring units)

〈uqtu−qt〉c 't→∞

2σvt

(q2 +m2)2. (75)

The q dependence is similar to the so-called Larkin random-force model [21], but with a time-dependent amplitude, i.e.the effective disorder is growing with time, which is naturalgiven the correlations (73). The correlation of the positionthus remains non-stationary at all times2.

From Eq. (75) one obtains the correlations of the displace-ment in real space, still in the large-t limit

(uxt − u0t)2' 2vt

∫ddq

(2π)d1

(q2 +m2)2(1− cos qx)

∼ vt× x2ζL (76)

with ζL = (4 − d)/2 the Larkin roughness exponent. Note

that the average displacement is uxt = vt − 1−e−m2t

m2 (seeAppendix J ). Hence we see that the BFM roughness scalingu ∼ x4−d is dimensionally consistent with the correlation atlarge times,

(uxt − u0t)2 ' 2 uxt x4−d . (77)

This result, ζ = 4 − d = ε, is in agreement with the FRGapproch: the position theory of the BFM model is an exactfixed point for the flow equation of the FRG with a roughnessexponent ζ = ε, as discussed in [10, 12].

2 Note that there are stationary versions of the BFM, which we will not dis-cuss here, see discussions in e.g. [11–13].

11

VII. CONCLUSION

We presented a general investigation of the Brownian ForceModel, using its exact solvability via the instanton equation invarious settings. After reviewing the results and the calcula-tions of [9, 11–13], we extended the study in several direc-tions.

First, we computed observables containing informationabout the spatial structure of avalanches in the BFM: the jointdensity of S and S0 (or equivalently, the distribution of thelocal size S0 at fixed total global size S), and the distributionof the extension ` of an avalanche. These distributions displaypower laws in their small-scale regime, which we recoveredusing scaling arguments, together with universal amplitudes.

We also extended the method to study new driving proto-cols relevant to distinct experimental setups. The derived re-sults show new exponents for the small-scale behavior of theglobal avalanche-size distribution following a locally imposeddisplacement, and for the small-scale behavior of the local-size distribution following a localized kick.

Finally, we presented results for the non-stationary dynam-ics of the BFM, focusing on observables which exist only inthe position theory, such as the roughness exponent. This ex-plains why both the Larkin roughness and the BFM roughness(emerging from the FRG approach), play a role in this model,depending on whether the driving is stationary or not.

ACKNOWLEDGMENTS

We thank A. Rosso, A. Kolton and A. Dobrinevski for stim-ulating discussions, PSL for support by Grant No. ANR-10-IDEX-0001-02-PSL, as well as KITP for hospitality and sup-port in part by NSF Grant No. NSF PHY11-25915.

Appendix A: Airy functions

We recall the definition of the Airy function:

Ai(z) :=

∫ ∞−∞

dt

2πeit3

3 +izt . (A1)

The following formula is usefulfor a ∈ R∗,

Φ(a, b, c) =

∫C

dz

2iπea

z3

3 +bz2+cz (A2)

= |a|−1/3e2b3

3a2− bca Ai

(b2

|a|4/3− c sgn(a)

|a|1/3

).

It can be obtained from (A1), deforming the contour C, e.g.to z = − b

a + iR.

- 0.5 0.5 1.0 1.5

- 0.4

- 0.2

0.2

0.4

0.6

0.8

1.0

FIG. 8. Representation of the potential energy V (y) as a function ofy, and lines of constant total energy, with E = 0 in red, E > 0 inblue and, E < 0 in green.

- 0.5 0.5 1.0 1.5

-1.0

- 0.5

0.5

1.0

FIG. 9. Phase-space diagram, i.e. trajectories represented with y′ asa function of y. The case E = 0 is in red, E > 0 in blue andand E < 0 in green. We can see that properties of the solution(periodicity, divergences, etc.) strongly depend on the value of E.

Appendix B: General considerations on the instanton equation

1. Sourceless equation

a. Massive case

It is useful to start with the simpler sourceless instantonequation

y′′ = y − y2 . (B1)

Here we denote by a prime the derivative with respect to x.It can be interpreted as the classical equation of motion ofa particule (of mass 2) in a potential V (y) = −y2 + 2y3

3 ,represented in Fig. 8. Multiplying by y′ and integrating once,we obtain y′ = ±

√E − V (y), where E is a real integration

constant equivalent to the total “energy” of the particle. Itsphase-space diagram (y, y′) is represented in Fig. 9.

From figures 8 and 9, we see that:

12

- 6 - 4 - 2 2 4 6

0.2

0.4

0.6

0.8

1.0

1.2

1.4

- 6 - 4 - 2 2 4 6

- 12

- 10

- 8

- 6

- 4

- 2

FIG. 10. Solutions with energy 0 of equation (B1); left : y+(x), right : y−(x).

(i) - there is exactly one positive E = 0 solution y+(x)defined for all x ∈ R, up to a shift x→ x+ x0. It reads∫ 3/2

y+(x)

dy√y2 − 2

3y3

= |x|

⇔ y+(x) =3

1 + coshx=

3

2

[1− tanh2

(x2

)].

(B2)

(ii) - There is exactly one negative E = 0 (zero energy)solution y−(x) defined for all x ∈ R∗, namely∫ y−(x)

−∞

dy√y2 − 2

3y3

= |x|

⇔ y−(x) =3

1− coshx=

3

2

[1− coth2

(x2

)].

(B3)

(iii) - There are two classes of solutions with E 6= 0. Thefirst class is defined on an interval of finite length r(E) with

r(E) = 2

∫ t

−∞

dy√E + y2 − 2

3y3

(B4)

where t 6= 0 denotes the smallest real root of E = −t2 + 23 t

3.This integral is convergent at large negative y due to the cubicterm, and also convergent near the root y = t (for E → 0 itdiverges logarithmically). If one chooses x = 0 as center ofthe interval, the solution y(x) satisfies∫ t

y(x)

dy√E + y2 − 2

3y3

= |x| . (B5)

It diverges at both ends x = ±r(E)/2. It is sometimes moreconvenient to choose x = 0 as the endpoint of the interval]0, r(E)[. Then, for x ∈]0, r(E)[ one has∫ y(x)

−∞

dy√E + y2 − 2

3y3

= x . (B6)

Setting y = 12 − z, this can be rewritten as

√6

∫ ∞12−y(x)

dz√4z3 − 3z + (1 + 6E)

= x . (B7)

This gives, in terms of the Weirstrass elliptic function P ,

y(x) =1

2− P

(x√6

; g2 = 3, g3 = −1− 6E

). (B8)

It diverges at x = 0 and x = r(E), and is the proper solutionon the interval ]0, r(E)[, see Appendix I.

The second class of solutions with E 6= 0 exists only for− 1

3 < E < 0; these solutions are periodic on the whole realline. As can be seen from Figs. 8 and 9, y(x) varies in abounded and strictly positive interval. We will not discussthese solutions as they will not be needed below.

b. Massless case

Consider now the massless sourceless equation,

y′′ = −y2 . (B9)

The analysis is similar to the massive case discussed abovewith V (y) = − 2

3y3. Its solutions have the following proper-

ties:(i) - there is no positive E = 0 solution.(ii) - There is only one negative E = 0 solution y−(x)

defined for all x ∈ R∗,∫ y−(x)

−∞

dy√− 2

3y3

= |x| ⇔ y−(x) = − 6

x2. (B10)

It can be obtained by considering the limit x 1 in the solu-tion (B3).

(iii) - There is now only one class of solutions with E 6= 0(the periodic ones have disappeared). They are defined on

13

an interval of length r(E). They have E = 23 t

3, hence t =

(3E/2)1/3 and

r(E) = 2

∫ t

−∞

dy√23 t

3 − 23y

3

=

√

6π(

23|E|

)1/6Γ(1/3)Γ(5/6) , E > 0

√6π(

23|E|

)1/62Γ(7/6)Γ(2/3) , E < 0 .

(B11)

The solution y(x) satisfies for x ∈]0, r(E)[∫ y(x)

−∞

dy√E − 2

3y3

= x . (B12)

It can be expressed in terms of the Weirstrass function,

y(x) = −P(x√6

; g2 = 0, g3 = −6E

). (B13)

It diverges at x = 0 and x = r(E). The periods are consistentwith

√6 × 2Ω (see Appendix I) using the relation Γ(7/6)

Γ(2/3) =Γ(1/3)3

4×21/3π3/2 . Note also the relation Γ(1/3)Γ(5/6) = 2×22/3π3/2

3Γ(2/3)3 .

2. Instanton solution with a single delta source

We now use these results to construct the solutions in pres-ence of sources. For a single delta source this was done in [9]and [13]. We first recall and then extend this analysis, as amore general approach is needed here.

a. Massive case

Consider the instanton equation

u′′(x)− u(x) + u(x)2 = −λδ(x) . (B14)

We are looking for a solution defined for all x ∈ R. Otherphysical requirements3 (e.g. from the derivation of the dynam-ical action) is that u(x) vanishes as x → ±∞, and that thesolution is analytic around λ = 0 (obtainable in a power se-ries in λ). We need a function which is piecewise solution ofEq. (B1) for x ∈]−∞, 0[ and for x ∈]0,∞[, with a disconti-nuity in its derivative,

u′(0+)− u′(0−) = −λ . (B15)

As we have seen in the previous section, in order to be definedon an infinite interval, it must be constructed from the zero-energy E = 0 solutions y±(x) of (B1) up to a shift x →

3 Because of finite range elasticity, the the effect at x = 0 of a kick at xmustdecay at large x. Because of the cutoff Sm, the positive integer momentsof avalanche sizes must exist

y+

y-

λ>0λ<0

x→-∞

x→+∞

x=0y'

y

FIG. 11. Graphical representation of the construction of solutions ofthe instanton equation for λ > 0 (blue) and λ < 0 (green). Thedotted part of the curve represents the discontinuity in the derivative.The red line represents the E = 0 solution of (B1), the only oneneeded to solve the instanton equation with one local source.

x+x0. By symmetry it reads u(x) = y±(|x|+x0) where x0 ≡x0(λ) is chosen to satisfy the condition (B15). The procedureis illustrated in Fig. 11. Note that the sign of λ dictates whichof the branches ± must be chosen. To summarize,

uλ(x) =3

1 + sλ cosh(|x|+ x0)=

3

2

[1− hλ(|x|+ x0)2

].

(B16)The function x0(λ) is determined from

λ =6sλ sinh(x0)[

1 + sλ cosh(x0)]2 =

3

2hλ(x0)

[1− hλ(x0)2

](B17)

with sλ = sgn(λ), hλ(x) = tanh(x2 ) for λ > 0 and hλ(x) =

coth(x2 ) for λ < 0. 4

This form does not make explicit that uλ(x) is analytic in λnear λ = 0. We will thus use the following equivalent form.Introduce z = hλ(x0). Equation (B17) can then be rewrittenas a cubic equation for z ≡ z(λ),

λ = 3z(1− z2) . (B18)

The trigonometric addition rules allow to rewrite

uλ(x) =3(1− z2)

2[

cosh(x2

)+ z sinh

(|x|2

) ]2=

6(1− z2)e−|x|[1 + z + (1− z)e−|x|

]2 .(B19)

The appropriate branch for (B18) is the one for which z → 1as λ → 0 (corresponding to x0 → ∞). As can be seen inFig. 12, this branch is defined for λ ∈]−∞, λc = 2√

3[, while

4 Note that formally x0 → x0 + iπ is equivalent to λ→ −λ.

14

x0= -x*

x0= x*

x0= -∞

x0= +∞

x0= 0y= y+

y= y-

FIG. 12. The generating function Z(λ) = 6(1 − z) is representedhere with some indications of the link with the construction of theinstanton solution; the green and blue dot correspond to the solutionsrepresented in figure 11.

z(λ) decreases from z(−∞) = ∞ to zc = z(λc) = 1/√

3.The other branches are solutions of (B14) but do not satisfythe physical requirements mentioned above.

Equations (B18) and (B19) thus define the solution to theinstanton equation for λ ∈] − ∞, λc[, in a way which is ex-plicitly analytic around λ = 0. For instance one can checkthat the small-λ expansion

uλ(x) =λ

2e−|x| +

λ2

6

(e−|x| − 1

2e−2|x|

)+O(λ3) (B20)

obtained by iteratively solving Eq. (B14) at small λ, is repro-duced by Eqs. (B18) and (B19).

Finally the partition sum corresponding to an homogeneouskick is expressed as

Z(λ) =

∫ ∞−∞dx uλ(x) = 6(1− z) . (B21)

Hence, from Eq. (B18), it satisfies

λ =1

72Z(Z − 6)(Z − 12) , (B22)

recovering the result obtained in [9].

b. Massless case

The massless instanton equation

u′′(x) + u(x)2 = −λδ(x) (B23)

is solved similarly. For λ < 0 there is a solution defined forall x ∈ R,

uλ(x) = − 6

(|x|+ x0)2, x3

0 = −24

λ. (B24)

Note that for the massless case the physical solution is notrequired to be analytic in λ at λ = 0 (i.e. integer moments ofavalanche sizes diverge). This solution can be obtained from(B19) in the (formal) double limit of small x and large z, withx0 = 2/z. The equation determining z now is λ = −3z3. Thegenerating function for a uniform kick becomes Z = −6z =(72λ)1/3.

Appendix C: Calculation of probabilities and densities of S0

For an arbitrary kick δwx, in the massive case, the Laplacetransform of the distribution of local size is∫

dS0 eλS0Pδwx(S0) = exp

(Ld−1

∫dx δwxu

λ(x)

).

(C1)Here uλ(x) is given in Eq. (B19). Performing the Laplaceinversion in general is difficult, but there are some tractablecases.

1. Uniform kick

Let us start with a uniform kick δwx = δw, and δw =Ldδw. It is more efficient to take a a derivative of Eq. (C1)w.r.t. λ and write the Laplace inversion for S0Pw(S0),

S0Pδw(S0) =

∫C

dλ

2iπe−λS0∂λe

6δw(1−z(λ)) . (C2)

HereC is an appropriate contour parallel to the imaginary axisand we used that

∫dx u(x) = 6(1− z). The function z(λ) is

solution of λ = 3z(1− z2). One can now use z as integrationvariable and rewrite

S0Pδw(S0) = 6δwe6δw

∫C

dz

2iπe−3z(1−z2)S0e−6δwz , (C3)

using dλ∂λ = dz∂z . We will be sloppy here about the in-tegration contour, as this procedure is heuristic to guess theresult, which will then be tested (see below). As the exponen-tial contains a cubic term, we use the Airy integral formula ofAppendix A leading to

S0Pw(S0) = 6δwe6δwΦ(a, b, c) . (C4)

Here Φ is defined in Eq. (A2), with a = 9S0, b = 0 andc = −(3S0 +6δw). This immediately leads to formula (21) inthe main text. We have checked numerically that it reproducesthe correct Laplace transform (C1) for λ < λc.

15

2. Local kick

For a local kick it is possible to calculate the PDF of thelocal jump at the position of the kick.

Consider a local kick at x = 0, i.e. δwx = δw0δ(x). Forsimplicity in this subsection we set d = 1. Inserting this valuein (C1) we find that the LT of the PDF of the local size at thesame point S0 reads∫

dS0 eλS0Pδw0

(S0) = e32 (1−z2)δw0 (C5)

using uλ(0) = 32 (1 − z2). The same manipulations as above

lead to

S0P (S0) = −∫C

dz

2iπe−3z(1−z2)S0∂ze

32 (1−z2)δw0 (C6)

= 3δw0e3δw0

2

∫C

dz

2iπz e−3z(1−z2)S0− 3

2 z2δw0

= 3δw0e3δw0

2 ∂cΦ(a, b, c)|a=9S0,b=− 3δw0

2 ,c=−3S0.

Using Eq. (A2) leads to

Pδw0(S0) =

δw0eδw0−

δw30

36S20

31/3S5/30

[δw0

2× 31/3S2/30

Ai(u)− Ai′(u)

]

u = 31/3S2/30 +

δw20

4× 32/3S4/30

. (C7)

We can check normalization, and that 〈S0〉 = 12δw0, consis-

tent with the small-λ expansion of (C5). The asymptotics are

Pδw0(S0) '

δw

3/20 e

δw02 −

δw30

18S20

√6πS2

0

for S0 1

δw0eδw0− 2√

3S0

2 4√

3√πS

3/20

for S0 1 .

(C8)

This result, and the new exponent τ = 5/3 of the divergenceat small S0, which appear when δw0 → 0, is discussed in themain text.

Appendix D: Calculation of the joint density of S and S0

We will obtain the joint density from the generating func-tion of S0 and S,

〈eλS0+µS〉 = e∫xδwxux (D1)

in terms of the solution of the instanton equation. Let us con-sider a uniform kick δwx = δw.

1. Instanton equation and its solution

a. Massive case

Here u (that we will also denote uλ,µ to make the depen-dence on the sources explicit) is the solution, in the variable

x, of the instanton equation

u′′ − u+ u2 = −λδ(x)− µ . (D2)

We must solve this equation with similar requirements as dis-cussed below for Eq. (B14), except that now the instantongoes to a constant at infinity (since the source acts every-where). Clearly, the new uniform source can be removed bya shift u → u + c, where the constant c verifies µ = c − c2.This results in the mass term −u → −(1 − 2c)u, which canbe brought back to Eq. (D2) with µ = 0, i.e. Eq. (B14), by asimple scale transformation. At the end one can check thatgiven uλ(x) the solution of Eq. (B14), the solution of Eq.(D2), noted uλ,µ(x), is given by

uλ,µ(x) =1− β2

2+ β2uλ/β

3

(βx) . (D3)

The constant β > 0 such that

β2 = β2µ :=

√1− 4µ . (D4)

In summary, the instanton solution is

uλ,µ(x) =1− β2

2+

6β2(1− z2)e−β|x|[1 + z + (1− z)e−β|x|

]2 , (D5)

where z is the solution of

λ

β3= 3z(1− z2) . (D6)

It is connected to z = 1 at λ = 0.

b. Massless case

It is useful to also give the solution in the massless case, forwhich one needs to solve

u′′ + u2 = −λδ(x)− µ (D7)

for µ ≤ 0. Using a shift and a rescaling we can check that thesolution now is

uλ,µ(x) =−β2

2+ β2uλ/β

3

(βx) . (D8)

The parameter β > 0 such that β2 =√−4µ, and uλ(x) is the

massive instanton solution. In summary, this gives

uλ,µ(x) =−β2

2+

6β2(1− z2)e−β|x|[1 + z + (1− z)e−β|x|

]2 (D9)

where z is again the solution (D6). If µ → 0, hence β → 0we recover the massless instanton (B24).

16

2. Joint distribution

Let us again consider the massive case. To obtain the jointprobability distribution Pδw(S, S0), we need to calculate thegenerating function Z(λ, µ),

〈eλS0+µS〉 =

∫ ∞0

∫ ∞0

Pδw(S0, S)eλS0+µSdS dS0

=eδwZ(λ,µ)

(D10)

Integrating (D5), we obtain

Z(λ, µ) =

∫x

uλ,µ(x) = Ld1− β2

2+ Ld−16βz(1− z)

=: LdZ1(µ) + Ld−1Z2 (λ, µ) .

(D11)

Z1(µ) is the generating function for the distribution of thetotal size of avalanches and Z2(λ, µ) a new term defined by(D11). The volume factors come from the coordinates alongwhich the instanton solution is constant.

From equations (D6) and (D11), we can express λ as a func-

tion of Z2 and β,

λ = 3β3

(1− Z2

6β

)[1−

(1− Z2

6β

)2]. (D12)

This is equivalent to Z2(λ, µ) = βZ(λβ3

)where Z ≡ Z(λ)

is the generating function of the local size, which was implic-itly defined as a solution of Eq. (B22).

Considering the limit of small δw, we obtain Pδw(S, S0) ≈δw ρ(S, S0), which defines the joint density ρ(S, S0) of totaland local sizes in the limit of a single avalanche. To simplifythe computation, we decompose the distribution ρ(S, S0) as

ρ(S, S0) = ρ(S, S0) + δ(S0) (ρ(S)− ρ(S))

ρ(S) =

∫S0>0

ρ(S, S0) . (D13)

Here ρ(S, S0) is the smooth part of the joint density for S andS0, and is also the joint density of single avalanches contain-ing 0 (i.e. S0 > 0). The second term takes into account allavalanches that occur away from 0: the δ(S0) ensures that theavalanche does not contain 0 and the subtraction ensure that∫S0ρ(S, S0) = ρ(S) where ρ(S) is the global size density. As

we will check at the end of the calculation, the correct gener-ating function for ρ is Z2(λ, µ)Ld−1 + 6 (1− βµ)Ld−1.

As ρ(S) is already known, we only want to computeρ(S, S0). To eliminate the term δ(S0) we multiply (D13) byS0 and use that S0ρ(S, S0) = S0ρ(S, S0). Multiplication byS0 is equivalent to taking a derivative w.r.t. λ in the generatingfunction,

S0 ρ(S0, S) = Ld−1

∫ i∞

−i∞

dµ

2πie−µS

∫ i∞

−i∞

dλ

2πie−λS0∂λZ2 (λ, µ)

= Ld−1

∫ i∞

−i∞

dµ

2πie−µS

∫ i∞

−i∞

dZ

2πie−

β3

72Zβ (6−Zβ )(12−Zβ )S0 .

(D14)

Here we changed variables from λ to Z2 (and dropped the indice) using (D12). To simplify the calculations, we introduce a newvariable x, s.t. Z = 2× 3

13x+ 6β, with β defined in Eq. (D4),

ρ(S0, S) = Ld−1 × 2× 313

S0

∫ i∞

−i∞

dµ

2πie−µS

∫ i∞

−i∞

dx

2πie−

x3

3 S0+31/3β2xS0

= Ld−1 × 2× 313e−S/4

S0

∫ i∞

−i∞

dx

2πie−

x3

3 S01

4

∫ i∞

−i∞

dy

2πie−

yS4 +(−y)1/231/3xS0

= Ld−1 × 2× 313e−S/4

S0

∫ i∞

−i∞

dx

2πie−

x3

3 S0

∫ ∞0

dy

4πe−

yS4 sin

(√y3

13xS0

)= Ld−1 × 2× 3

23

e−S/4√πS

32S0

∫ i∞

−i∞

dx

2πie−

x3

3 S0xS0e− (31/3xS0)2

S

(D15)

The steps of this calculations are: first a linear change of vari-able 4µ − 1 → y, such that β = (−y)

12 , then a deformation

of the contour of integration to integrate on both sides of the

branch cut R+. Finally, the last integration can be performed

17

in terms of Airy functions (e.g. using Appendix A),

ρ(S, S0) =6Ld−1

√πS2

e−S4 F(√

3S0/S34

)(D16)

F (u) =1

u23

e−23u

4(u

43 Ai

(u

83

)− Ai′

(u

83

)).

The density of avalanches with global size S and which con-tain 0, i.e. with S0 > 0 is

ρ(S) =

∫ ∞0

dS0 ρ(S, S0)

= Ld−1 × 2

√3

π

[ ∫ ∞0

duF (u)

]e−

S4

S54

= Ld−1 3 Γ(

14

)2π

e−S4

S54

(D17)

where

3 Γ(

14

)2π

= 2

√3

π

∫ ∞0

duF (u) ≈ 1.7311012158 . (D18)

To test our solution one can check that∫ ∞0

ds3 Γ(

14

)2π

e−S4

S54

(eµS−1) = 6[1− (1− 4µ)

14

]. (D19)

We have checked numerically several other requirements,originating from the definitions, namely∫ ∞

0

dS ρ(S, S0) = ρ0(S0) =2Ld−1

πS0K1/3

(2S0/

√3)

∫ ∞0

dS

∫ ∞0

dS0 S0ρ(S, S0) = Ld−1∫ ∞0

dS

∫ ∞0

dS0 Sρ(S, S0) = 6Ld−1∫ ∞0

dS

∫ ∞0

dS0 ρ(S, S0)eµS(eλS0 − 1) = Z2(λ, µ)Ld−1 .

Appendix E: Imposed local displacement

We set for simplicity d = 1 in this section. The PDF of theglobal size in presence of imposed position driving is obtainedfrom

eµS = em2ux=0δw , (E1)

where ux is the solution of a slightly modified instanton equa-tion:

u′′x −m2δ(x)ux + u2x = −µ (E2)

and we have kept explicit the local mass. This equation is thesame as the massless Eq. (D7), with λ = −m2ux=0, a self-consistency condition. Using its solution given in Eqs. (D9)and (D6) we eliminate λ and z in the system λ = −m2ux=0 = −m2β2

(1− 3

2z2)

λ

β3= 3z(1− z2)

(E3)

- 1.0 - 0.5 0.5 1.0 1.5 2.0

- 200

- 150

- 100

- 50

FIG. 13. Instanton solutions involved in the computation of Z(r) forr = 1: in blue, u1(x), in red u∞(x) and in purple, u∞(x− 1).

with β = (−4µ)1/4. It is then easy to see that there is asolution such that m2ux=0 remains finite when m2 → ∞, inwhich case z →

√23 and

limm2→∞

m2ux=0 = −√

2

3β3 . (E4)

Hence we find

Pδw0(S) = LT−1−µ→S e

−δw√

23 (−4µ)3/4 . (E5)

The result for the density is simpler,

Sρ(S) = −LT−1−µ→S ∂µ

√2

3(−4µ)3/4 , (E6)

leading to

ρ(S) =

√3

Γ(1/4)S7/4(E7)

and a new exponent 7/4 discussed in the main text.

Appendix F: Some elliptic integrals for the extensiondistribution

Here we make explicit the calculation for the density of ex-tensions sketched in the main text. The relevant generatingfunction, defined in the main text in Eq. (67), is

Z(r) =

∫x

ur(x)− u∞(x)− u∞(x− r) . (F1)

Here ur(x) is the solution of the instanton equation with twolocal sources, one at x = 0 and one at x = r. The solutionu∞ with one source at x = 0 and one at infinity is equivalentto the solution with only one source at x = 0.

The first simplification in the calculation of this integral isthe symmetry arround r/2. Another is that, for x ∈]−∞, 0[,ur(x) − u∞(x) cancels exactly. Then, the idea is to express

18

the integral for Z(r) without explicitly solving the instantonequation, using the change of variables∫

u dx =

∫udu

u′. (F2)

This requires to express the derivative of u w.r.t. x as a func-tion of u, which is easy because u is solution of a differentialequation, and to decompose the integral into two parts suchthat the change of variables is well defined: from x = −∞ tox = 0 and from x = 0 to x = r/2. The rest is deduced bysymmetry.

In these two intervals, u∞(x − r) does not contain a pole,and can safely be computed separately. Moreover, as we said,ur(x) − u∞(x) vanishes in the first interval, i.e. for x ∈] −∞, 0]. This leaves only the integral of ur(x) − u∞(x) overx running from x = 0 to x = r/2. To simplify notations weintroduce the variable t < 0,

t := ur(r/2) , (F3)

which is in one-to-one correspondance with r, and is a nice

parameter to express Z. Indeed, after the change of variables(F2), the integral now runs from u = −∞ to u = t, and for0 < x < r/2, with u ≡ ur, we have

u′r =

√−t2 +

2

3t3 + u2 − 2

3u3 . (F4)

Further, with u ≡ u∞,

u′∞ =

√u2 − 2

3u3 . (F5)

This comes from the results of Appendix B, and the relationE = −t2 + 2

3 t3. To express r in terms of t, we use the same

idea as in the derivation of Eq. (F2),

r = 2

∫ r/2

0

dx = 2

∫ t

−∞

duru′r

. (F6)

Putting these ingredients together, we obtain Z(r) as a func-tion of t, which we call Z(t), in term of an elliptic integral, aswell as the expression of r as a function of t,

Z(t) = 2

∫ t

−∞

u√−t2 + 2

3 t3 + u2 − 2

3u3− u√

u2 − 23u

3

du− 2

∫ 0

t

u√u2 − 2

3u3du

= 2t

∫ ∞1

y√y2 − 1− 2

3 t(y3 − 1)

− 1√1− 2

3 ty

dy − 6 + 2√

9− 6t ,

(F7)

r(t) = 2

∫ t

−∞

du√−t2 + 2

3 t3 + u2 − 2

3u3

= 2

∫ ∞1

dy√y2 − 1− 2

3 t(y3 − 1)

. (F8)

We now use this to characterise the small-size divergence ofthe extension distribution. This is encoded in the small r be-havior of Z(r), which corresponds to the large-t behavior ofZ(t). For the latter, we have

Z(t) ' −2

√3

2

[∫ ∞1

du

(u√

u3 − 1− 1√

u

)− 2

]|t| 12

' 2√

6πΓ(5/6)

Γ(1/3)|t| 12 ,

(F9)

which is also the exact result in the massless limit. We nextneed to invert Eq. (F8) in the large-t limit,

|t| ' A2r−2 , A = 2√

6πΓ(7/6)

Γ(2/3)=√

6Γ(1/3)3

42/3π. (F10)

The small-r behavior of Z(r) is then given by

Z(r) ' 4√

3π r−1 . (F11)

For small |t| we find

r(t) ' 2 ln(12/|t|) (F12)

and

Z(t) ' t2 ln(1/|t|) (F13)

which leads to

Z(r) = 72 re−r +O(e−r) (F14)

This leads to the tail of the extension density,

ρ(`) = ∂2r Z(r)

∣∣∣r=`' 72 ` e−` when `→∞ . (F15)

Appendix G: Joint distribution for extension and total size

For simplicity, we consider only m = 0 (massless limit).To obtain the joint distribution of extension and total size wehave to add a global source µ to the instanton equation, in ad-dition to the two local sources, whose parameters are sent toinfinity. With the same tricks as previously, c.f. Appendix Dand notably Eq. (D8), we change this problem to a new one

19

with a mass β = (−4µ)14 , but no global source. The generat-

ing function is now a function of r, the distance between thetwo local sources and β, the new mass. As in Appendix F, wecan change the variable r to the new parameter t defined inEq. (F3) and express everything in terms of elliptic integrals:

r(t, β) = 2

∫ ∞t

dy√−β2t2 − 2

3 t3 + β2y2 + 2

3y3

= β−1f

(t

β2

),

Z(t, β) = −2

∫ ∞t

y√−β2t2 − 2

3 t3 + β2y2 + 2

3y3− 1√

23y

+ 2√

6t = β g

(t

β2

).

(G1)

The functions f and g are

f(x) = 2

∫ ∞x

du√−x2 − 2

3x3 + u2 + 2

3u3

= 2

∫ ∞1

du√u2 − 1 + 2

3x(u3 − 1),

g(x) = −2x

∫ ∞1

u√u2 − 1 + 2

3x(u3 − 1)− 1√

23xu

+ 2√

6x .

(G2)

From that, we have Z(r, β) = β(g f−1

)(βr) and then

∂2rZ(r, β) = β3

(g f−1

)′′(βr) which gives

ρ(l, s) =1

4l7F

(sl−4

4

), (G3)

where F is the inverse LT of x 7→ (−x)34 (gf−1)′′

((−x)

14

)with g and f the functions previously defined. Giving an an-alytic expression for this scaling function F seems not to bepossible.

Appendix H: Numerics

We test most of our results with a direct numerical simula-tion of the equation of motion (1). This is done by discretiz-ing both time and space. To avoid the

√dt term from a naive

Euler time discretisation, we use the method of [22], whichallows to express the exact propagator of the d = 0 versionof (1) in terms of random distributions (Poisson and Gammadistribution). We here review this result.

Let us start with the d = 0 stochastic equation,

∂tut = α− βut +√

2σut η(t) (H1)

where η is a Gaussian white noise and α is positive (so that uremains non-negative at all times). It can be integrated exactlyusing Bessel functions (cf. [12] for a derivation of this usingthe instanton equation for the ABBM model):

P (ut|u0) =β

σ

√utu0

−1+α

2 sinh(βt2

)I−1+α

βσ

√utu0

sinh(βt2

)(e βt2 )α e− βσ u0e−βt+ut1−e−βt . (H2)

To use this representation efficiently in a numerical algorithm, the trick is to expand it in a series, and then express it as acombination of two distributions,

P (ut|u0) =

∞∑n=0

un−1+αt un0

n!Γ(n+ α)

β

2σ sinh(βt2

)2n+α (

eβt2

)αe− βσ

u0eβt−1 e

− βσut

1−e−βt

=

∞∑n=0

Poisson[β

σ

u0

eβt − 1

](n) Gamma

[n+ α,

1− e−βt

βσ

](ut) .

(H3)

20

The Poisson and Gamma distributions used above are

Poisson [λ] (n) = e−λλn

n!for n ∈ N (H4)

Gamma [k, θ] (x) =1

θ(k − 1)!

(xθ

)k−1

e−xθ for x ∈ R

(H5)

This means that we can generate ut at time t from u0 bychoosing first n according to the Poisson distribution and thenchoosing ut from a Gamma distribution with a shape depend-ing on n. This can be summed up as a nice equality betweenrandom variables,

ut = Gamma[

Poisson[β

σ

u0

eβt − 1

]+ α,

1− e−βt

βσ

](H6)

To use this in a numerical simulation of Eq. (1), we first writea discretized (in space) version of the latter,

∂tui,t = (ui+1,t + ui−1,t)− (m2 + 2)ui,t +√

2σui,tξi,t

+m2δwi,t . (H7)

Choosing α = ui+1,t + ui−1,t, which is assumed to be con-stant on the time interval [t, t+dt], and β = m2+2 in Eq. (H5)allows us to generate ui,t+dt, knowing all ui,t, with a correctprobability distribution at order dt.

Appendix I: Weierstrass and Elliptic functions

Here we recall some properties of Weierstrass’s ellipticfunction P (source [23] chapter 18, and Wolfram Mathworld).It appears in complex analysis as the only doubly periodicfunction on the complex plane with a double pole 1/z2 atzero5. Denoting ω1, ω2 the two (a priori complex) primi-tive half-periods, every point of the lattice Λ = 2mω1 +2nω2|(n,m) ∈ Z2 is a pole of order 2 for P . It can be con-structed for z ∈ C− Λ as

P(z|ω1, ω2) :=1

z2

+∑

m,n6=(0,0)

1

(z − 2mω1 − 2nω2)2− 1

(2mω1 + 2nω2)2.

(I1)

It is an even function of the complex variable z, with P(z) =P(−z). Note that the choice of primitive vectors (2ω1, 2ω2)is not unique, since one can alternatively choose any linearcombination. The conventional choice of roots g2 and g3 isdefined from its expansion around z = 0,

P(z|ω1, ω2) =1

z2+g2

20z2 +

g3

28z4 +O(z6) . (I2)

5 It also appears as the second derivative of the Green function of the freefield on a torus.

The function P is alternatively denoted

P(z|ω1, ω2) = P(z; g2, g3) (I3)

the latter being defined in Mathematica asWeierstrassP[z, g2, g3]. More explicitly, the parame-ters g2, g3 are expressed from the half-periods as

g2 = 60∑

m,n6=(0,0)

1

(2mω1 + 2nω2)4, (I4)

g3 = 140∑

m,n 6=(0,0)

1

(2mω1 + 2nω2)6. (I5)

The Weierstrass elliptic function verifies an interesting homo-geneity property,

P(λz;λ−4g2, λ−6g3) = λ−2P(z; g2, g3) , (I6)

and the non-linear differential equation

P ′(z)2 = 4P(z)3 − g2P(z)− g3 . (I7)

It is thus linked to elliptic integrals. Restricting now tog2, g3 ∈ R and focusing on z ∈ R one can choose onehalf-period to be real, which we denote Ω 6. The functionP(z) is then periodic in R of period 2Ω and diverges at allpoints 2mΩ, m ∈ Z. It is defined in the fundamental inter-val ]0, 2Ω[, repeated by periodicity. In this interval it satisfiesthe symmetry P(2Ω − z; g2, g3) = P(z; g2, g3). Its valuesin the first half-interval, i.e. for z ∈ [0,Ω] are such that (withy ∈ [e1,∞])

z =

∫ ∞y

dt√4t3 − g2t− g3

⇔ y = P(z; g2, g3) (I8)

where e1 is the largest real root of the polynomial in t

4t3 − g2t− g3 = 4(t− e1)(t− e2)(t− e3) . (I9)

The roots ei are all real if ∆ = g32 − 27g2

3 > 0 and only one,namely e1, is real if ∆ < 0. Hence the period is given by

Ω =

∫ ∞e1

dt√4t3 − g2t− g3

, P(Ω) = e1 , P ′(Ω) = 0 .

(I10)It is always finite, except when e1 is a double root, in whichcase ∆ = 0 and the period is infinite Ω =∞.

For g2 = 0 the integral (I10) can be calculated explicitlyusing∫ ∞

1

du√u3 − 1

=Γ(1/3)3

423π

=−√π Γ(1/6)

Γ(−1/3),∫ ∞

−1

du√u3 + 1

=√π

Γ(1/3)

Γ(5/6).

(I11)

6 The conventions are such that if ∆ < 0, Ω = ω1 is real and ω2 imaginary(for g3 > 0 and the reverse for g3 < 0), and if ∆ < 0, Ω = ω1 ± ω2.

21

The half-periods are

Ω =

1

4πΓ(1/3)3g

−1/63 when g3 > 0

√π

Γ(1/3)

413 Γ(5/6)

|g3|−1/6 when g3 < 0, (I12)

and the other period can be chosen as 12Ω(1 + i

√3).

Finally, taking another derivative of (I7) we see that theWeierstrass function also satisfies

P ′′(z) = 6P(z)2 − g2

2, (I13)

and P(z; g2, g3) is the only solution of this differential equa-tion which satisfies (I2).

From this we can find solutions of the instanton equation

u′′x −Aux + u2x = 0 , (I14)

whereA = 1 is the massive case andA = 0 the massless case.Comparing with Eq. (I13) we see that a family of solutions are

ux =A

2− 6b2P

(c+ bx;

A2

12b4, g3

). (I15)

Because of the homogeneity relation (I6), this is a two-parameter family. These solutions are periodic. In the mass-less case A = 0, the period of (I15) is Ω/b where Ω is givenby (I12).

Appendix J: Non-stationary dynamics

In the velocity theory the observables of the BFM are cal-culated from the dynamical action

S[u, u] =

∫t,q

u−q,t(∂t + q2 +m2)uq,t − σ∫t,x

u2xtuxt

where u is the response field. The quadratic part of the action,S0, defines the free response function,

〈uq,tuq,t′〉S0 := Rq,t−t′ = θ(t− t′)e−(q2+m2)(t−t′) . (J1)

Standard perturbation theory in the disorder σ is then per-formed, and has the peculiarity to contain only tree diagrams.It is easy to see that the average velocity is not corrected by thedisorder, hence its value is the same as in the free theory. Inpresence of a uniform drivingw = vt, and taking into accountthe initial condition uxt=0 = 0, one has

ux,t = 〈uxt〉S = v(

1− e−m2t). (J2)

This implies

uxt = vt− 1− e−m2t

m2. (J3)

Next we compute the connected correlations, where q meansFourier space and x real space,

uq,t1 u−q,t2c

= 〈uq,t1 u−q,t2〉S

= σ

∫s,x

〈uq,t1 u−q,t2 u2x,sux,s〉S0 (J4)

= 2σ

∫s

〈uxs〉S0Rq,t1−sRq,t2−s .

Calculating this integral, and further integrating over t1 and t2we obtain

uq,tu−q,tc =

∫ t

0

dt1

∫ t

0

dt2 uq,t1 u−q,t2c. (J5)

This is the final result given in the main text, see Eq. (74).Alternatively we can obtain the correlations of uxt using

eµxuxt1 =

∫x

µxUxt1 +1

2

∫x1x2

µx1µx2

Ux1t1Ux2t2

c+ ...

= exp

(vm2

∫x,t>0

uλxt

)(J6)

where uλxt is the solution of the space-time dependent instan-ton equation with a source λxt = µxθ(t)θ(t1 − t). Usingthe perturbation method in the source of Section III.H of [13],specializing to that source in (261), we obtain at the end thesame result as above.

[1] S. Zapperi, P. Cizeau, G. Durin and H.E. Stanley, Dynamics of aferromagnetic domain wall: Avalanches, depinning transition,and the Barkhausen effect, Phys. Rev. B 58 (1998) 6353–6366.

[2] P. Le Doussal, K.J. Wiese, S. Moulinet and E. Rolley, Heightfluctuations of a contact line: A direct measurement of therenormalized disorder correlator, EPL 87 (2009) 56001,arXiv:0904.1123.

[3] D.S. Fisher, Collective transport in random media: From su-perconductors to earthquakes, Phys. Rep. 301 (1998) 113–150.

[4] D. Bonamy, S. Santucci and L. Ponson, Crackling dynamicsin material failure as the signature of a self-organized dynamicphase transition, Phys. Rev. Lett. 101 (2008) 045501.

[5] S. Papanikolaou, F. Bohn, R.L. Sommer, G. Durin, S. Zapperiand J.P. Sethna, Universality beyond power laws and the aver-age avalanche shape, Nature Physics 7 (2011) 316–320.

[6] K. Dahmen and JP. Sethna, Hysteresis, avalanches, anddisorder-induced critical scaling: A renormalization-group ap-proach, Phys. Rev. B 53 (1996) 14872–14905.

22

[7] T. Nattermann, S. Stepanow, L.-H. Tang and H. Leschhorn, Dy-namics of interface depinning in a disordered medium, J. Phys.II (France) 2 (1992) 1483–8.

[8] O. Narayan and D.S. Fisher, Threshold critical dynamics ofdriven interfaces in random media, Phys. Rev. B 48 (1993)7030–42.

[9] P. Le Doussal and K.J. Wiese, Size distributions of shocks andstatic avalanches from the functional renormalization group,Phys. Rev. E 79 (2009) 051106, arXiv:0812.1893.

[10] P. Le Doussal and K.J. Wiese, First-principle derivation ofstatic avalanche-size distribution, Phys. Rev. E 85 (2011)061102, arXiv:1111.3172.

[11] P. Le Doussal and K.J. Wiese, Distribution of velocities in anavalanche, EPL 97 (2012) 46004, arXiv:1104.2629.

[12] A. Dobrinevski, P. Le Doussal and K.J. Wiese, Non-stationarydynamics of the Alessandro-Beatrice-Bertotti-Montorsi model,Phys. Rev. E 85 (2012) 031105, arXiv:1112.6307.

[13] P. Le Doussal and K.J. Wiese, Avalanche dynamics of elasticinterfaces, Phys. Rev. E 88 (2013) 022106, arXiv:1302.4316.

[14] A. Dobrinevski, P. Le Doussal and K.J. Wiese, Avalancheshape and exponents beyond mean-field theory, EPL 108 (2014)66002, arXiv:1407.7353.

[15] B. Alessandro, C. Beatrice, G. Bertotti and A. Montorsi,Domain-wall dynamics and Barkhausen effect in metallic fer-romagnetic materials. I. Theory, J. Appl. Phys. 68 (1990) 2901.

[16] B. Alessandro, C. Beatrice, G. Bertotti and A. Montorsi,Domain-wall dynamics and Barkhausen effect in metallic fer-romagnetic materials. II. Experiments, Journal of AppliedPhysics 68 (1990) 2908.

[17] F. Colaiori, Exactly solvable model of avalanches dynamics forbarkhausen crackling noise, Advances in Physics 57 (2008)287, arXiv:0902.3173.

[18] AA. Middleton, Asymptotic uniqueness of the sliding state forcharge-density waves, Phys. Rev. Lett. 68 (1992) 670–673.

[19] T. Thiery, P. Le Doussal and K.J. Wiese, Spatial shape ofavalanches in the Brownian force model, J. Stat. Mech. 2015(2015) P08019, arXiv:1504.05342.

[20] A. Dobrinevski, Field theory of disordered systems –avalanches of an elastic interface in a random medium,arXiv:1312.7156 (2013).

[21] A.I. Larkin, Sov. Phys. JETP 31 (1970) 784.[22] I. Dornic, H. Chate and M.A. Munoz, Integration of Langevin

equations with multiplicative noise and the viability of fieldtheories for absorbing phase transitions, Phys. Rev. Lett. 94(2005) 100601.

[23] M. Abramowitz and A. Stegun, Pocketbook of MathematicalFunctions, Harri-Deutsch-Verlag, 1984.

CONTENTS

I. Introduction 1

II. Avalanche observables of the BFM 2A. The Brownian Force Model 2B. Avalanches observables and scaling 2C. Generating functions and instanton equation 4

III. Distribution of avalanche size 4

A. Global size 4B. Local size 5C. Joint global and local size 5D. Scaling exponents 6

IV. Driving at a point: avalanche sizes 7A. Imposed local force 7B. Imposed displacement at a point 7

V. Distribution of avalanche extensions 7A. Scaling arguments for the distribution of

extensions 7B. Instanton equation for two local sizes 8C. Avalanche extension with a local kick 8D. Avalanche extension with a uniform kick 9

VI. Non-stationnary dynamics in the BFM 10

VII. Conclusion 11

Acknowledgments 11

A. Airy functions 11

B. General considerations on the instanton equation 111. Sourceless equation 11

a. Massive case 11b. Massless case 12

2. Instanton solution with a single delta source 13a. Massive case 13b. Massless case 14

C. Calculation of probabilities and densities of S0 141. Uniform kick 142. Local kick 15

D. Calculation of the joint density of S and S0 151. Instanton equation and its solution 15

a. Massive case 15b. Massless case 15

2. Joint distribution 16

E. Imposed local displacement 17

F. Some elliptic integrals for the extension distribution 17

G. Joint distribution for extension and total size 18

H. Numerics 19

I. Weierstrass and Elliptic functions 20

J. Non-stationary dynamics 21

References 21